well, assuming there is a finite number of such 3mod4 prime numbers, you can add them all together, multiply by 4 and add 3... and you get another 3mod4 prime, thus a contradiction
I only checked like 3 examples but I think that works, only a suggestion, im terrible with proofs

First none of the primes pj divides N since pj | N+1, so if we had pj|N then we get pj|(N+1)-N = 1, which is a contradiction.

Now alteast 1 of the prime factors of N has form 3 mod 4. Now N is odd, so if such a prime doesnt exist then all prime factors of N have form p = 1 mod 4, which contradicts the whole construction of N.